Multi-regime mixing modeling for local extinction and re-ignition in turbulent non-premixed flame by using LES/FDF method

Abstract

Local extinction and re-ignition occur in turbulent non-premixed combustion when the Damköhler number is not large enough and the combustion is not fully mixing controlled. The occurrence of local extinction introduces locally extinguished flame holes followed by a premixed flame propagation toward the hole center to potentially reignite the flame. The co-existence of the non-premixed and premixed combustion regimes complicates the modeling since traditional combustion models are mostly for a single regime. In this work, we examine the effect of multi-regime mixing modeling in the transported filtered density function (FDF) method on the predictions of local extinction and reignition. Predictions of local extinction and reignition remain a challenge for the FDF method despite the progress made in the past. To account for the multi-regime combustion, two different mixing timescale models for non-premixed and premixed combustion are combined. A flame index based on the gradients of fuel and oxidizer is used to define a weighting factor to blend the two mixing timescale models. A turbulent jet non-premixed flame with substantial local extinction, the Sydney piloted jet flame L, is adopted as a test case to examine the performance of the multi-regime model in large-eddy simulation/FDF modeling. It is found that the traditional non-premixed mixing timescale model when combined with the modified Curl mixing leads to global extinction for the Sydney flame L without the presence of the premixed combustion regime. After accounting for the multi-regime combustion with proper detection of the different combustion regimes, the predictions for the flame statistics and the amount of local extinction are significantly improved. It suggests the need of including multi-regime combustion for the predictions of local extinction and re-ignition in turbulent non-premixed combustion configurations.

Appendices

Appendix 1: Definition of burning index BI

The definition of the burning index BI in Xu and Pope (2000) is followed in this work,

$$\begin{aligned} \textrm{BI}(T)=\frac{\left\langle \rho T|\xi _l<\xi<\xi _u\right\rangle }{\left\langle \rho |\xi _l<\xi <\xi _u\right\rangle }\cdot \frac{1}{T_{r}}, \end{aligned}$$

(11)

where $\rho$ is density, T is temperature, $\xi$ is mixture fraction, $\langle \cdot |\cdot \rangle$ is the conditional average, $\xi _l$ and $\xi _u$ define the lower and upper limit of mixture fraction for the calculation of the conditional average, and $T_r$ is a reference temperature. The burning index based on other scalars like the species mass fractions can be defined similarly. In this work, only the burning index based on temperature is used. The experimental data for T and species mass fractions are used for the calculation of density and BI. The mixture fraction interval used to calculate the conditional average is chosen to be around the stoichiometric condition $\xi _{st}$. Specifically, the interval is $(\xi _l,\xi _u)=(0.301,0.401)$ for the Sandia flames ($\xi _{st}=0.351$) and $(\xi _l,\xi _u)=(0.035,0.095)$ for the Sydney flames ($\xi _{st}=0.055$). The reference value $T_r$ is taken to be the equilibrium temperature of the methane/air mixture at the stoichiometric condition, $T_r=2230.8$ K. This choice of $T_r$ is different from Xu and Pope (2000) where the peak value of temperature in a strained opposed jet laminar flame with a strain rate $a=100\,s^{-1}$ was used. The peak values of temperature in the opposed jet laminar flames based on the fuel/air configurations in the Sandia flames and Sydney flames are slightly different. We hence chose the equilibrium temperature which is the same for both flames to provide the same baseline for a direct comparison of the calculated flame index.

Fig. 9

Comparison of the burning index $\textrm{BI}(T)$ based on the measured temperature T against the axial distance x/D in the Sandia piloted flames E and F and the Sydney flame L

The calculated flame indices in the Sandia piloted flames E and F and the Sydney flame L are compared in Fig. 9. As illustrated in the figure, the computed burning index BI in the Sydney flame L is even lower than the Sandia flame F based on the experimental data. The lower value of BI in flame L than in flame F which is close to global extinction indicates that flame L has severe local extinction and hence is a challenging case for the model development and rigorous testing.

Appendix 2: Sensitivity to the grid and turbulence resolution scale

LES/PDF simulations of flame L with three different grids, $144\times 108\times 48$, $256\times 108\times 48$, and $512\times 192\times 96$, are conducted to examine the effect of different grids on the simulation results. In the simulations, the turbulence resolution scale $\mathrm {\Delta }$ (e.g., in Eq. (4)) is the same as the grid size. The grid effect thus also represents the effect of the turbulence resolution scale $\mathrm {\Delta }$ on the simulation results.

Fig. 10

Radial profiles of the time-averaged axial velocity $\langle \tilde{u}\rangle /U_\textrm{b}$, axial velocity r.m.s. $\langle \widetilde{u''^2}\rangle ^{1/2}/U_\textrm{b}$, mixture fraction $\langle \tilde{\xi }\rangle$, mixture fraction r.m.s. $\langle \widetilde{\xi ''^2}\rangle ^{1/2}$ predicted by using LES/FDF with three different grids, $144\times 108\times 48$ (solid lines), $256\times 108\times 48$ (dashed lines), and $512\times 192\times 96$ (dash-dotted lines) at the axial locations $x/D=$10, 20, and 30 in the Sydney flame L

Figure 10 compares the effect of the different grids on the predictions of turbulence and mixing in flame L. The radial profiles of the axial velocity $\langle \tilde{u}\rangle$ are captured qualitatively well by all three grids. The difference between the different grid results is insignificant, especially when comparing the grid $256\times 108\times 48$ that is adopted in the main study with the fine grid $512\times 192\times 96$. For the prediction of the axial velocity r.m.s. $\langle \widetilde{u''^2}\rangle ^{1/2}$, the coarse grid $144\times 108\times 48$ yields some over-prediction when compared with the fine grid and the experimental data. The middle grid $256\times 108\times 48$ produces reasonably accurate predictions with much lower computational cost when compared with the finest grid. For the mixing fields $\langle \tilde{\xi }\rangle$ and $\langle \widetilde{\xi ''^2}\rangle ^{1/2}$, the different grids do not yield much difference in the predictions.

Fig. 11

Radial profiles of the time-averaged residual shear stress $\langle \widetilde{u''v''}\rangle _\textrm{r}/U_\textrm{b}^2$ and resolved shear stress $\langle \widetilde{u''v''}\rangle _\textrm{R}/U_\textrm{b}^2$ predicted by using LES/FDF with three different grids, $144\times 108\times 48$ (solid lines), $256\times 108\times 48$ (dashed lines), and $512\times 192\times 96$ (dash-dotted lines) at the axial locations $x/D=$10, 20, and 30 in the Sydney flame L

Figure 11 examines the effect of the different grids (turbulence resolution scales) on the predictions of the residual shear stress $\langle \widetilde{u''v''}\rangle _\textrm{r}$ and the resolved shear stress $\langle \widetilde{u''v''}\rangle _\textrm{R}$ in flame L, where the subscripts “r" and “R" denote the residual scale and resolved scale in LES, respectively. All three grids yield similar predictions of the residual and resolved shear stresses with the predicted resolved shear stress one order of magnitude higher than the residual shear stress. To quantitatively estimate the level of turbulence resolution, we define a shear-stress resolution factor $\gamma$,

$$\begin{aligned} {\gamma =\frac{\max \left( \langle \widetilde{u''v''}\rangle _\textrm{R}\right) }{\max \left( \langle \widetilde{u''v''}\rangle _\textrm{R}\right) +\max \left( \langle \widetilde{u''v''}\rangle _\textrm{r}\right) },} \end{aligned}$$

(12)

where the maximum is taken along the radial direction at any given axial location in flame L. The resolution factor can be computed from the different grid cases at different axial locations. The results show that all the three grids resolved more than 82% of the shear stress ($\gamma >82\%$) at $x/D=10$ and more than 90% ($\gamma >90\%$) at $x/D=20$ and 30. This is consistent with the Pope criterion (Pope 2004) to resolve 80% of the turbulent kinetic energy to be a well-resolved LES.

Overall, the three examined grids yield similar predictions of the turbulence and mixing fields in flame L, all in reasonable agreement with the experimental data. The grid resolution is sufficient to resolve more than 80% of the shear stress. The intermediate grid $256\times 108\times 48$ adopted in the work is deemed to be a suitable choice for the study of the mixing and combustion in the Sydney flame L.